Simon, R. ; Mukunda, N. ; Dutta, Biswadeb (1994) Quantumnoise matrix for multimode systems: U(n) invariance, squeezing, and normal forms Physical Review A, 49 (3). pp. 15671583. ISSN 10502947

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Official URL: http://pra.aps.org/abstract/PRA/v49/i3/p1567_1
Related URL: http://dx.doi.org/10.1103/PhysRevA.49.1567
Abstract
We present a complete analysis of variance matrices and quadrature squeezing for arbitrary states of quantum systems with any finite number of degrees of freedom. Basic to our analysis is the recognition of the crucial role played by the real symplectic group Sp(2n,openR) of linear canonical transformations on n pairs of canonical variables. We exploit the transformation properties of variance (noise) matrices under symplectic transformations to express the uncertaintyprinciple restrictions on a general variance matrix in several equivalent forms, each of which is manifestly symplectic invariant. These restrictions go beyond the classically adequate reality, symmetry, and positivity conditions. Towards developing a squeezing criterion for nmode systems, we distinguish between photonnumberconserving passive linear optical systems and active ones. The former correspond to elements in the maximal compact U(n) subgroup of Sp(2n,openR), the latter to noncompact elements outside U(n). Based on this distinction, we motivate and state a U(n)invariant squeezing criterion applicable to any state of an nmode system, and explore alternative ways of expressing it. The set of all possible quantummechanical variance matrices is shown to contain several interesting subsets or subfamilies, whose definitions are related to the fact that a general variance matrix is not diagonalizable within U(n). Definitions, characterizations, and canonical forms for variance matrices in these subfamilies, as well as general ones, and their squeezing nature, are established. It is shown that all conceivable variance matrices can be generated through squeezed thermal states of the nmode system and their symplectic transforms. Our formulas are developed in both the real and the complex forms for variance matrices, and ways to pass between them are given.
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